private Diophantine Chakravala(Diophantine d, long N)
{
// see aforementioned wikipedia page for how this works. Essentially if you have two solutions to a more general
// a^2 - N*b^2 = k, you can compose the two solutions to find yet another. Eventually k == 1 and you have
// the solution we are looking for.
while (d.k != 1)
{
BigInteger m, a, b, k;
if (d.k == -1)
{
//special case, compose d with itself
a = (d.a * d.a + N * d.b * d.b);
b = (2 * d.a * d.b);
k = 1;
}
else
{
m = ChooseM(d, N);
a = (d.a * m + N * d.b) / BigInteger.Abs(d.k);
b = (d.a + d.b * m) / BigInteger.Abs(d.k);
k = (m * m - N) / d.k;
}
d.a = a;
d.b = b;
d.k = k;
}
return(d);
}
private BigInteger ChooseM(Diophantine d, long N)
{
// we need m such that k | (a + bm) that minimizes (m^2 - N)
// first find possible m
// a + bm = 0 mod k
// => m = -a(b^-1) mod k
var sqrtN = (int)Math.Sqrt(N);
var k = BigInteger.Abs(d.k);
var invB = Modular.ModularMultiplicativeInverse(d.b, k); // apparently gcd(b, k) = 1 always, so this is legit ¯\_(ツ)_/¯
var m = (-d.a) * invB;
var m0 = m + ((sqrtN - m) / k) * k;
var m1 = m + ((sqrtN - m) / k + 1) * k;
return(new[] { m0, m1 }.OrderBy(x => BigInteger.Abs(x * x - N)).First());
}
public static void diophantine_solution_minimize_test()
//****************************************************************************80
//
// Purpose:
//
// DIOPHANTINE_SOLUTION_MINIMIZE_TEST tests DIOPHANTINE_SOLUTION_MINIMIZE.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 02 December 2006
//
// Author:
//
// John Burkardt
//
{
const int a = 4096;
const int b = -15625;
const int c = 46116;
Console.WriteLine("");
Console.WriteLine("DIOPHANTINE_SOLUTION_MINIMIZE_TEST");
Console.WriteLine(" DIOPHANTINE_SOLUTION_MINIMIZE computes a minimal");
Console.WriteLine(" Euclidean norm solution of a Diophantine equation:");
Console.WriteLine(" A * X + B * Y = C");
int x = 665499996;
int y = 174456828;
int r = a * x + b * y - c;
Console.WriteLine("");
Console.WriteLine(" Coefficients:");
Console.WriteLine(" A = " + a.ToString().PadLeft(12) + "");
Console.WriteLine(" B = " + b.ToString().PadLeft(12) + "");
Console.WriteLine(" C = " + c.ToString().PadLeft(12) + "");
Console.WriteLine(" Solution:");
Console.WriteLine(" X = " + x.ToString().PadLeft(12) + "");
Console.WriteLine(" Y = " + y.ToString().PadLeft(12) + "");
Console.WriteLine(" Residual R = A * X + B * Y - C:");
Console.WriteLine(" R = " + r.ToString().PadLeft(12) + "");
Diophantine.diophantine_solution_minimize(a, b, ref x, ref y);
r = a * x + b * y - c;
Console.WriteLine("");
Console.WriteLine(" DIOPHANTINE_SOLUTION_MINIMIZE returns");
Console.WriteLine(" the minimized solution:");
Console.WriteLine(" X = " + x.ToString().PadLeft(12) + "");
Console.WriteLine(" Y = " + y.ToString().PadLeft(12) + "");
Console.WriteLine(" Residual R = A * X + B * Y - C:");
Console.WriteLine(" R = " + r.ToString().PadLeft(12) + "");
x = 15621;
y = 4092;
r = a * x + b * y - c;
Console.WriteLine("");
Console.WriteLine(" Here is the minimal positive solution:");
Console.WriteLine(" X = " + x.ToString().PadLeft(12) + "");
Console.WriteLine(" Y = " + y.ToString().PadLeft(12) + "");
Console.WriteLine(" Residual R = A * X + B * Y - C:");
Console.WriteLine(" R = " + r.ToString().PadLeft(12) + "");
}
public static void diophantine_test()
//****************************************************************************80
//
// Purpose:
//
// DIOPHANTINE_TEST tests DIOPHANTINE.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 02 December 2006
//
// Author:
//
// John Burkardt
//
{
const int TEST_NUM = 20;
int[] a_test =
{
1027, 1027, 1027, 1027, -1027,
-1027, -1027, -1027, 6, 0,
0, 0, 1, 1, 1,
1024, 0, 0, 5, 2
};
int[] b_test =
{
712, 712, -712, -712, 712,
712, -712, -712, 8, 0,
1, 1, 0, 0, 1,
-15625, 0, 3, 0, 4
};
int[] c_test =
{
7, -7, 7, -7, 7,
-7, 7, -7, 50, 0,
0, 1, 0, 1, 0,
11529, 1, 11, 19, 7
};
bool error = false;
int test_i;
int x = 0;
int y = 0;
Console.WriteLine("");
Console.WriteLine("DIOPHANTINE_TEST");
Console.WriteLine(" DIOPHANTINE solves a Diophantine equation:");
Console.WriteLine(" A * X + B * Y = C");
Console.WriteLine("");
Console.WriteLine(" A B C X Y Residual");
Console.WriteLine("");
for (test_i = 0; test_i < TEST_NUM; test_i++)
{
int a = a_test[test_i];
int b = b_test[test_i];
int c = c_test[test_i];
Diophantine.diophantine(a, b, c, ref error, ref x, ref y);
switch (error)
{
case true:
Console.WriteLine(a.ToString().PadLeft(10) + " "
+ b.ToString().PadLeft(10) + " "
+ c.ToString().PadLeft(10) + " "
+ "(Error occurred!)" + "");
break;
default:
int r = a * x + b * y - c;
Console.WriteLine(a.ToString().PadLeft(10) + " "
+ b.ToString().PadLeft(10) + " "
+ c.ToString().PadLeft(10) + " "
+ x.ToString().PadLeft(10) + " "
+ y.ToString().PadLeft(10) + " "
+ r.ToString().PadLeft(10) + "");
break;
}
}
}
